Telephone TraflSc Time Averages 



By JOHN RIORDAN 



(Manuscript Received April 25, 1951) 



This paper describes the determination of the first four semi-invariants of the 

 distribution of the average, over an arbitrary time interval, of traffic carried by a 

 telephone system with an infinite number of trunks, during a period of statistical 

 equilibrium. Both finite and infinite numbers of independent call sources are con- 

 sidered, and the distribution function of call holding times is left general. 



1. Introduction 



T?OR mathematical studies of telephone traffic, like those of call loss or 

 ^ delay which are used in trunking engineering, the traffic is considered 

 as a flow of probability in time. In the period of most importance, the busy 

 hour, this flow is usually regarded as stationary; that is to say, the proba- 

 bility of a given number of busy trunks, or the probability of delay of an 

 incoming call (or any other probability of the system which comes in ques- 

 tion) is taken as independent of the particular moment in the busy hour at 

 which the system is examined. The system is said to be in statistical equilib- 

 rium. 



For such theoretical studies, the statistical quantities which determine 

 these probabilities, like the rate at which calls appear, are of course taken 

 as given, but in the application they must be determined by observations, 

 such as those being taken in the current extensive program of traffic meas- 

 urements. Here a difficulty appears. To abridge the extensive amount of 

 observational material, either measurements are made of traffic averages 

 over periods small compared to the busy hour (but not small enough to be 

 neglected) or the measurements of continuous recorders are averaged by 

 hand. It may be noticed here that for application of the results given below 

 the traffic averages obtained by measurements must be those of a con- 

 tinuous device which records all traffic changes and not, as in some measuring 

 devices, those obtained from a number of ''looks" at points within the 

 averaging interval. But to use these measurements in determining the 

 traffic parameters by standard sampling theory, a corresponding theoretical 

 study of the averages is necessary. 



Such a study, within limits to be described presently, is given here. No 

 attempt is made to describe the sampling studies possible from the results 

 reached. These seem to be of many kinds, not necessary to describe, but for 



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